Problem: Solve for $x$ : $ 6|x - 10| - 1 = -4|x - 10| + 8 $
Add $ {4|x - 10|} $ to both sides: $ \begin{eqnarray} 6|x - 10| - 1 &=& -4|x - 10| + 8 \\ \\ { + 4|x - 10|} && { + 4|x - 10|} \\ \\ 10|x - 10| - 1 &=& 8 \end{eqnarray} $ Add ${1}$ to both sides: $ \begin{eqnarray} 10|x - 10| - 1 &=& 8 \\ \\ { + 1} &=& { + 1} \\ \\ 10|x - 10| &=& 9 \end{eqnarray} $ Divide both sides by ${10}$ $ \dfrac{10|x - 10|} {{10}} = \dfrac{9} {{10}} $ Simplify: $ |x - 10| = \dfrac{9}{10}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -\dfrac{9}{10} $ or $ x - 10 = \dfrac{9}{10} $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -\dfrac{9}{10} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -\dfrac{9}{10} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -\dfrac{9}{10} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $10$ $ x = - \dfrac{9}{10} {+ \dfrac{100}{10}} $ $ x = \dfrac{91}{10} $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = \dfrac{9}{10} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& \dfrac{9}{10} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& \dfrac{9}{10} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $10$ $ x = \dfrac{9}{10} {+ \dfrac{100}{10}} $ $ x = \dfrac{109}{10} $ Thus, the correct answer is $x = \dfrac{91}{10} $ or $x = \dfrac{109}{10} $.